Question

If \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \end{bmatrix} \), then \( A^2 \) is equal to

• A. unit matrix
• B. null matrix
• C. A
• D. -A

Hint:

Matrices of this form often satisfy \(A^2 = I\) (involutory matrix).

Answer 1

The Correct Option is A

Concept: Compute \(A^2 = A \cdot A\) using matrix multiplication.

Step 1: Set up multiplication. \[ A^2 = \begin{bmatrix} 1 & 0 & 0 0 & 1 & 0 a & b & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 0 & 1 & 0 a & b & -1 \end{bmatrix} \]

Step 2: First two rows. \[ R_1: (1,0,0) \to (1,0,0) \] \[ R_2: (0,1,0) \to (0,1,0) \]

Step 3: Third row. \[ (a,b,-1) \begin{bmatrix}10a\end{bmatrix} = a - a = 0 \] \[ (a,b,-1) \begin{bmatrix}01b\end{bmatrix} = b - b = 0 \] \[ (a,b,-1) \begin{bmatrix}00-1\end{bmatrix} = 1 \]

Step 4: Final matrix. \[ A^2 = \begin{bmatrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{bmatrix} = I \]